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<b>let </b><font color="Maroon" title="c1">Y</font>, <font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">Y</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X</font> implies <font color="Maroon" title="c1">Y</font> is   <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c2">X</font> )</span><br/>



<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c1">Y</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">Y</font> is   <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c2">X</font></span><br/>

<a NAME="E2:31"/>
<font color="Maroon" title="c1">Y</font> is <a href="finsub_1.html#V2" title="FINSUB_1:attr.2">cap-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="finsub_1.html#D2" title="FINSUB_1:def.2">FINSUB_1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> or  not <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> or <font color="Maroon" title="c3">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> )</span><br/>

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">A'</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c6">B'</font> = <font color="Maroon" title="c4">B</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">A1</font> = <font color="Maroon" title="c5">A'</font> <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <font color="Maroon" title="c6">B'</font>;<br/>
<a NAME="E3:31_1"/><i><font color="Green" title="E22">A5</font></i>: 
 <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A'</font> <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <font color="Maroon" title="c6">B'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T30" target="_self" title="PROB_1:th.30">Th30</a></i>;<br/>
<a NAME="E4:31_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A'</font> <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <font color="Maroon" title="c6">B'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c6">B'</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T128" title="FUNCT_7:th.128">FUNCT_7:128</a></i>;<br/>
<a NAME="E5:31_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A'</font> <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <font color="Maroon" title="c6">B'</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="txt" href="#E1:31_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/>
<b>hence </b><a NAME="E6:31_1"/>
<font color="Maroon" title="c3">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E3:31_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="prob_1.html#D8" target="_self" title="PROB_1:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:31"/>
<font color="Maroon" title="c1">Y</font> is   <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E20">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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